As the skies sobbed a heavy rain down on Lower Manhattan yesterday, most rescue workers toiling through the debris at Ground Zero stubbornly refused to give up hope of finding survivors, even as the number of people missing soared to 6,333.

“We’re trying our best to keep morale up. We’re all a little frustrated that we haven’t been able to find anyone,” said a haggard Officer Bob Schnelle of the NYPD’s K-9 unit. “But we’re going to keep at it until they tell us to stop.”

Mayor Giuliani said that won’t happen until after the disaster site is completely cleared.

“It’ll never, ever be a situation in which we’re not sensitive to and looking to see if there are people that are alive there,” the mayor told NBC’s “Today” show.

But, he added, families should be girding themselves for the possibility that the remains of some of the victims may never be found.

The mayor said the sudden jump in the number of missing yesterday – over 900 more people than were reported the day before – came from foreign nationals who were in the World Trade Center complex.

While yesterday’s rainfall was a slight help in dousing intermittent fires, it remained more of a hindrance.

The water made debris heavier and the dust muddy, slowing down the rescuers.

The amount of time that’s passed since any survivors were found – eight days – is taking a toll on some of the searchers.

Firefighter Jerry Cox, of Engine Co. 263 in Astoria, said he’s now looking for “something to give someone closure, whether it be a pocketbook or, unfortunately, body parts. With any luck, you get whole bodies out, and let families have something to bury.”

Police Commissioner Bernard Kerik, meanwhile, said he’s worried about the psychological impact the gruesome discoveries are having on his rank and file, and instructed his commanders to help them with “the emotional issues” – and to watch for officers who might be nearing “overload.”

Still, Kerik said, it’s the strength of his officers that has kept him going during these trying times on just three hours of sleep a night.